2 Answers
2

I'll refer to my ancient book "Simplicial objects in algebraic topology". It is best to restrict to Kan complexes $K$ with a single vertex. In 23.3 and 23.4, it is shown that the path projection $PK \to K$ is a particularly nice kind of simplicial bundle provided that its
fiber $L(K)$ is a simplicial group, which usually fails. The Kan loop group (Section 26)
$G(K)$ substitutes for $L(K)$. It is the fiber of a different simplicial bundle over $K$ with a contractible total space. The geometric realization of this bundle is equivalent to the path space fibration of the realization $|K|$.